BIZOMAT MAKEOS · Strength
Column Strength by Rankine–Gordon
What load will this column carry by the Rankine–Gordon formula?
Crippling load of a column by the Rankine–Gordon formula, which blends crushing and Euler buckling, with the theoretical or an empirical Rankine constant and the safety factor.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Crushing (compressive) strength | MPa | Crushing or compressive yield strength of the material. |
| Elastic modulus | GPa | Elastic modulus from the material data sheet (steel is typically about 200 GPa). |
| Area | mm² | Cross-section area. |
| Least second moment of area | mm⁴ | Least second moment of area. |
| Length | mm | Unsupported length. |
| Effective length factor | Effective length factor for the end conditions (1.0 pinned–pinned). | |
| Rankine constant a (0 = theoretical Sc ÷ π²E) | Empirical Rankine constant for pinned ends from your reference (e.g. about 1/7500 for mild steel); 0 uses the theoretical Sc ÷ π²E. | |
| Applied load | N | Applied compressive load. |
| Required safety factor | Factor of safety required by your design code, customer specification or company standard. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Crushing (compressive) strength
- What it is
- The stress at which a short, stocky piece of the material fails in compression by crushing or compressive yield. It sets the load a very short column can carry.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 320 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from the material standard or supplier data; for ductile metals it is usually the compressive yield strength.
What this means: Elastic modulus
- What it is
- Young’s modulus of the column material. It is used only when the Rankine constant is left at 0, to work out the theoretical constant.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter GPa, not MPa. It has no effect if you enter an empirical Rankine constant.
What this means: Area
- What it is
- The cross-sectional area of the column, from a section table, CAD or the section properties tools.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0.000001 mm² to 100000000 mm². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the gross area of the actual section, not the area of the outline.
What this means: Least second moment of area
- What it is
- The smallest second moment of area of the section, about the axis the column will buckle around most easily.
- Units and format
- Enter a number in mm⁴: millimetres to the fourth power (second moment of area). The form accepts 0.000001 mm⁴ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400000 mm⁴ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the least (minor axis) value; using the major axis overestimates the load.
What this means: Length
- What it is
- The length of column between the points where it is held against sideways movement, before the effective length factor is applied.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure between lateral restraints, which may be shorter than the overall length.
What this means: Effective length factor
- What it is
- The multiplier that turns the actual length into an effective buckling length for the end conditions, so that KL is the length between points of zero moment.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Slenderness rises in proportion to K, which lowers the Rankine load.
- Get it right
- Choose it from your design code for the real end fixity; pinned–pinned ends are 1.
What this means: Rankine constant a (0 = theoretical Sc ÷ π²E)
- What it is
- The Rankine constant that sets how quickly the column load falls with slenderness. Enter an empirical value from your reference for the material, or 0 to use the theoretical value Sc ÷ π²E.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger constant lowers the Rankine load for the same slenderness.
- Get it right
- Empirical constants are usually quoted as a fraction for pinned ends; type the decimal, not the denominator.
What this means: Applied load
- What it is
- The compressive load the column actually carries in service, the working load before any safety factor.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safety factor is the Rankine load divided by this load.
- Get it right
- Use the axial load along the column; eccentric loading needs a separate bending check.
What this means: Required safety factor
- What it is
- The minimum ratio of Rankine load to applied load that the column must reach for the result to be marked adequate.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set by your design code, customer or company standard; it changes only the verdict, not the load.
Method and formulas
- r = √(I/A); slenderness λ = KL/r.
- P_cr = Sc A ÷ (1 + a λ²), with a = Sc ÷ π²E (theoretical) or an empirical value.
- Safety factor = P_cr ÷ P.
- Example: the default strut has λ = 100 and a Rankine load of 122.08 kN.
Limitations
- Empirical blend of crushing and buckling; codes (e.g. AS 4100) use other column curves.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check columns and struts of any slenderness with one formula
- Compare with Euler and Johnson results
- Use published Rankine constants for cast iron or timber
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Slenderness ratio
- Rankine constant used
- Crippling load and safety factor
Interpreting the result
- Compare with the Euler and Johnson tool; the Rankine result lies below both limits.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 281–287. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
When yield meets buckling
What it covers: Why Euler's formula overestimates stocky members, and how the Johnson parabola, the Rankine–Gordon formula and classic column formulas bridge the gap between squashing and buckling.
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